Math Visualizations

What is an Integral?

An integral accumulates quantities over an interval. If $r(t)$ is a rate of change, then the integral gives the total accumulated amount:

$$R(t) = \int_0^t r(\tau) \, d\tau$$

Rainfall Accumulation

In this visualization, $r(t) = 5 - 0.5t$ represents rainfall rate (decreasing over time). The accumulated rainfall is:

$$R(t) = \int_0^t (5 - 0.5\tau) \, d\tau = 5t - 0.25t^2$$

• The rate graph (top) shows how fast rain falls at each moment
• The shaded area under the rate curve equals the total accumulated rainfall
• The accumulation graph (bottom) shows the running total

Riemann Sums

The area under the curve can be approximated by rectangles. Each rectangle has width $\Delta t$ and height $r(t_i)$:

$$\int_0^t r(\tau) \, d\tau \approx \sum_{i=1}^{n} r(t_i) \, \Delta t$$

Toggle the Riemann rectangles to see how they approximate the area. As the number of rectangles increases, the approximation improves.

Try This

1. Drag the slider to change the integration endpoint
2. Watch the shaded area grow as you extend the interval
3. Toggle Riemann rectangles to see the approximation
4. Compare the accumulated value with the area under the rate curve

Real-Life Applications

• Water tank filling — rate of flow integrated over time gives total volume
• Electricity bills — power usage rate integrated over time gives total energy
• Drug dosage — medication absorption rate integrated gives total drug in body